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Linear Function






Linear Function

Linear Function is defined as the real valued function $f : R \rightarrow R$ , y = f(x) = mx + c for each $x \in R$ where m and c is a constant
Linear function

$y = x+ 2$
$y=2x -3$
$y=x-2$
$y =4x$
The above all are example of linear function

Domain and Range of the Linear Function


For $f : R \rightarrow R$ , y = f(x) = mx + c for each $x \in R$

Domain = R
Range = R


Graph of the Linear Function

We can draw the graph on the Cartesian plan with value of x on the x-axis and value of y=f(x) on the y-axis. We can plot the point and join the point to obtain the graph. Here in case of the constant function,the graph will be a straight line parallel to x axis
For $y = f(x) = mx + c$ \(m\) is the slope and \(c\) is the \(y\) intercept of the graph.
If \(m\) is positive
then the line rises to the right and if \(m\) is negative then the line falls to the right
Below are few graph based on values of m and c
Graph for the linear function with m and c both positive
Graph of Linear function with positive slope and intercept
Graph for the linear function with positive m and negative c
Graph of Linear function with positive slope and negative intercept
Graph for the linear function with m and c both negative
Graph of Linear function with negative slope and intercept
Graph for the linear function with negative m and positive c
Graph of Linear function with negative slope and positive intercept
Graph for the linear function with positive m and c=0. This passes through origin. Such type of linear function is also called proportional function
Graph of Linear function with positive slope and zero intercept
Graph for the linear function with negative m and c=0. This passes through origin
Graph of Linear function with negative slope and zero intercept

Identify Function and constant function are special cases of Linear function
if m =1 and c=0, Linear function becomes f(x) =x which is a identity function
If m=0 ,then Linear function becomes f(x) =c which is a Constant function

Solved examples of Linear Functions

1. which is below function is a Linear function?
a. $y =2x$
b. $y = 11 -x$
c. $ y= \frac {2}{3} x + \frac {1}{4} $
d. $ x^2 + y^2=1$
e. $y =x^3$
f. $y =x^2 +1$
Solution
For the function to be a Linear function ,it should be of the form (mx+c)
a. This is Linear function as of the form (mx+c)
b. This is Linear function as of the form (mx+c)
c. This is Linear function as of the form (mx+c)
d. This is not a linear function
e. This is not a linear function
f. This is not a linear function

2. which of the graph represent Linear function?
example of linear function graph
Solution
The graph should be straight line for the function to be constant function
So C and D are constant function

3. Let f = {(1,1), (2,3), (0, -1), (-1, -3)} be a linear function from Z into Z.
Find f(x).
Solution
Since f is a linear function, f (x) = mx + c. Also, since $(1, 1), (0, - 1) \in Function$,
f (1) = m + c = 1 and f (0) = c = -1. This gives m = 2 and so,f(x) = 2x - 1.

Quiz Time

Question 1The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by
$t(C) =\frac {9}{5} C + 32$
Which of the following is incorrect?
A.t(0)=32
B. t(-5)=23
C. t(10)=48
D. None of the above
Question 2If a function is defined as $f = {(x, y) | y = 2x + 7, \; where \; x \in R \; and \; -5 \leq x \leq 5}$ is a relation. Then find the domain and Range of Function?
A. Domain=[-5,5], range=[3,17]
B. Domain=[-5,5], range=[-3,17]
C. Domain=[-5,5], range=[-3,-17]
D. Domain=[-5,5], range=[-5,5]
Question 3 Let f = {(1,1), (2,3), (3,5), (4,7)} be a linear function from Z into Z.
and f(x) =px +q then
A. p=1,q=1
B. p=1,q=2
C. p=2,q=1
D. p=2,q=-1
Question 4 Let $f(x) =c $,find the value of $f(2) -f(1)$
A. 4
B. 2
C. 0
D. 1
Question 5 The slope of the Linear function y=11x-1 is
A. 0
B. 11
C. -1
D. None of these
Question 6Find the Range and domain of the function $f(x) =x +2$
A. Domain = R, Range =R
B. Domain = R - {2}, Range = R
C. Domain = R , Range = R -{2}
D. Domain = R - {1}, Range = R



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